Homework 1

Practice with for loops

The purpose of this section is to give you some more practice working with for loops and sequences, which are useful tools for efficiently repeating a process many times. Here is an example for loop that calculates \(\sqrt{x}\) for a sequence of numbers \(x = 0, 0.1, 0.2, ..., 0.9, 1\):

x <- seq(from=0, to=1, by=0.1)
sqrt_x <- rep(0, length(x))
for(i in 1:length(x)){
  sqrt_x[i] <- sqrt(x[i])
}
sqrt_x

Below are some short practice questions to help you get more comfortable creating for loops.

Note: In Questions 1 and 2, you are applying a function to each element in a vector. Here you have used a for loop, because the purpose of these questions is to practice loops. However, for loops are not always the most efficient way to write code. Instead, many functions in R are vectorized: if you apply the function to a vector, it is applied to each element of the vector. For example,

x <- seq(from=0, to=1, by=0.1)
sqrt_x <- sqrt(x)
sqrt_x

produces the same output as the for loop above.

Probability simulation

Consider a tug-of-war competition for robots. In each match up, two robots take turns tugging the rope until the marker indicates that one of the robots won. The match starts with the marker at 0.

• Robot A pulls the rope – use runif(n=1,min=0,max=0.50) to simulate the magnitude of the pull. Adding the simulated value to the marker position gives the new position of the marker.
• Robot B pulls the rope in the opposite direction – use runif(n=1,min=0,max=0.50) to simulate the magnitude of the pull. Adding the simulated value to the marker position gives the new position of the marker.
• The two robots continue taking turns until the marker moves past -0.50 or 0.50.

Designing simulation studies

In class, we have started to discuss the use of simulation studies to address statistical questions, such as “How important is the normality assumption in a simple linear regression model?” A simulation study allows us investigate these questions by simulating data under a variety of different conditions (e.g., different violations of the normality assumption), and seeing how the statistical methods behave under these different conditions.

The paper “Using simulation studies to evaluate statistical methods” (Morris et al. 2019) provides a good overview of the important steps in designing a simulation study. Read sections 1 (Introduction) and 3 (Planning simulation studies), and then answer the following questions.

A new simulation study

We return here to the simple linear regression model:

\[Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i\] Another assumption from STA 112 is that the noise term \(\varepsilon_i\) has constant variance; that is, \(Var(\varepsilon_i) = \sigma^2\) for all observations (the variance does not depend on \(X_i\), e.g.).

Suppose we want to design a simulation study to assess how important the constant variance assumption is. In this section of the assignment, you will plan out your simulation study. In HW 2, you will carry out the simulations.

In class, we used the following code to simulate data from a simple linear regression model with normal errors:

n <- 100
beta0 <- 0.5
beta1 <- 1
x <- runif(n, min=0, max=1)
noise <- rnorm(n, mean=0, sd=1)
y <- beta0 + beta1*x + noise

Notice that the errors here also have constant variance: sd = 1 for all errors in the simulation. However, for our new simulation study, we will need to simulate data for which the standard deviation is different for different observations. For example, in the simulation above we could set \(SD(\varepsilon_i) = X_i\), or \(SD(\varepsilon_i) = X_i^2\).